Let $a_n,\ n \geq 0$ be complex numbers such that $\lim_n a_n = 0$.\\
(A) Show that $F(z) := \sum_{n \geq 0} a_n z^n$ is a holomorphic function on $\{z \in \mathbb{C} : |z| < 1\}$.\\
(B) Let $G(z)$ be a meromorphic function on $\{z \in \mathbb{C} : |z| < 2\}$, with a pole at 1. Show that $G \neq F$ on $\{z \in \mathbb{C} : |z| < 1\}$. (Hint: consider the function $(1-z)F(z)$ as $z \longrightarrow 1$.)